Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit. Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point.

Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.

t(hours)R(t)(gallons per hour)01113823510

The rate at which water leaks from a container is modeled by the twice-differentiable function R, where R(t) is measured in gallons per hour and t is measured in hours for 0≤t≤1. Values of R(t) are given in the table above for selected values of t.
(a) Use the data in the table to find an approximation for R′(12) . Show the computations that lead to your answer. Indicate units of measure.
(b)Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate ∫10R(t)ⅆt . Indicate units of measure.
(c) Use the data in the table to evaluate ∫130R′(t)ⅆt.
d) The sum ∑k=1nR(14+k2n)12n is a right Riemann sum with n subintervals of equal length. The limit of this sum as n goes to infinity can be interpreted as a definite integral. Express the limit as a definite integral.